3.739 \(\int x^{-4+n} (a+b x)^{-n} \, dx\)

Optimal. Leaf size=110 \[ -\frac{2 b^2 x^{n-1} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}+\frac{2 b x^{n-2} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac{x^{n-3} (a+b x)^{1-n}}{a (3-n)} \]

[Out]

-((x^(-3 + n)*(a + b*x)^(1 - n))/(a*(3 - n))) + (2*b*x^(-2 + n)*(a + b*x)^(1 - n))/(a^2*(2 - n)*(3 - n)) - (2*
b^2*x^(-1 + n)*(a + b*x)^(1 - n))/(a^3*(1 - n)*(2 - n)*(3 - n))

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Rubi [A]  time = 0.0356771, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{2 b^2 x^{n-1} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}+\frac{2 b x^{n-2} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac{x^{n-3} (a+b x)^{1-n}}{a (3-n)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-4 + n)/(a + b*x)^n,x]

[Out]

-((x^(-3 + n)*(a + b*x)^(1 - n))/(a*(3 - n))) + (2*b*x^(-2 + n)*(a + b*x)^(1 - n))/(a^2*(2 - n)*(3 - n)) - (2*
b^2*x^(-1 + n)*(a + b*x)^(1 - n))/(a^3*(1 - n)*(2 - n)*(3 - n))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int x^{-4+n} (a+b x)^{-n} \, dx &=-\frac{x^{-3+n} (a+b x)^{1-n}}{a (3-n)}-\frac{(2 b) \int x^{-3+n} (a+b x)^{-n} \, dx}{a (3-n)}\\ &=-\frac{x^{-3+n} (a+b x)^{1-n}}{a (3-n)}+\frac{2 b x^{-2+n} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}+\frac{\left (2 b^2\right ) \int x^{-2+n} (a+b x)^{-n} \, dx}{a^2 (2-n) (3-n)}\\ &=-\frac{x^{-3+n} (a+b x)^{1-n}}{a (3-n)}+\frac{2 b x^{-2+n} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac{2 b^2 x^{-1+n} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}\\ \end{align*}

Mathematica [A]  time = 0.025607, size = 64, normalized size = 0.58 \[ \frac{x^{n-3} (a+b x)^{1-n} \left (a^2 \left (n^2-3 n+2\right )+2 a b (n-1) x+2 b^2 x^2\right )}{a^3 (n-3) (n-2) (n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-4 + n)/(a + b*x)^n,x]

[Out]

(x^(-3 + n)*(a + b*x)^(1 - n)*(a^2*(2 - 3*n + n^2) + 2*a*b*(-1 + n)*x + 2*b^2*x^2))/(a^3*(-3 + n)*(-2 + n)*(-1
+ n))

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Maple [A]  time = 0.007, size = 77, normalized size = 0.7 \begin{align*}{\frac{ \left ( bx+a \right ){x}^{-3+n} \left ({a}^{2}{n}^{2}+2\,abnx+2\,{b}^{2}{x}^{2}-3\,{a}^{2}n-2\,abx+2\,{a}^{2} \right ) }{ \left ( bx+a \right ) ^{n} \left ( -3+n \right ) \left ( -2+n \right ) \left ( -1+n \right ){a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-4+n)/((b*x+a)^n),x)

[Out]

(b*x+a)*x^(-3+n)*(a^2*n^2+2*a*b*n*x+2*b^2*x^2-3*a^2*n-2*a*b*x+2*a^2)/((b*x+a)^n)/(-3+n)/(-2+n)/(-1+n)/a^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-4+n)/((b*x+a)^n),x, algorithm="maxima")

[Out]

integrate(x^(n - 4)/(b*x + a)^n, x)

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Fricas [A]  time = 1.62168, size = 208, normalized size = 1.89 \begin{align*} \frac{{\left (2 \, a b^{2} n x^{3} + 2 \, b^{3} x^{4} +{\left (a^{2} b n^{2} - a^{2} b n\right )} x^{2} +{\left (a^{3} n^{2} - 3 \, a^{3} n + 2 \, a^{3}\right )} x\right )} x^{n - 4}}{{\left (a^{3} n^{3} - 6 \, a^{3} n^{2} + 11 \, a^{3} n - 6 \, a^{3}\right )}{\left (b x + a\right )}^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-4+n)/((b*x+a)^n),x, algorithm="fricas")

[Out]

(2*a*b^2*n*x^3 + 2*b^3*x^4 + (a^2*b*n^2 - a^2*b*n)*x^2 + (a^3*n^2 - 3*a^3*n + 2*a^3)*x)*x^(n - 4)/((a^3*n^3 -
6*a^3*n^2 + 11*a^3*n - 6*a^3)*(b*x + a)^n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-4+n)/((b*x+a)**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-4+n)/((b*x+a)^n),x, algorithm="giac")

[Out]

integrate(x^(n - 4)/(b*x + a)^n, x)